MATH2088 Number Theory and Cryptography Prerequisites:(MATH1014 or MATH1002 or MATH1902) and (MATH1001 or MATH1901 or MATH1906 or MATH1003 or MATH1903 or MATH1907 or MATH1011 or MATH1021 or MATH1921 or MATH1931 or MATH1013 or MATH1023 or MATH1923 or MATH1933 or MATH1008 or MATH1904 or MATH1064)</p>

18/12/2018

Unit of study

Credit points

A: Assumed knowledge P: Prerequisites C: Corequisites N: Prohibition

Session

Mathematics

For a major in Mathematics, the minimum requirement is 24 credit points from senior units of study listed in this subject area.

Junior units of study

Introductory level

MATH1111Introduction to Calculus

6

A Knowledge of algebra and trigonometry equivalent to NSW Year 10 N MATH1011 or MATH1901 or MATH1906 or MATH1001 or HSC Mathematics Extension 1 or HSC Mathematics Extension 2 or ENVX1001 or MATH1021 or MATH1921 or MATH1931

Note: Department permission required for enrolmentStudents who have previously successfully studied calculus at a level at least equivalent to HSC Mathematics are prohibited.

Semester 1

Fundamental level

MATH1011Applications of Calculus

3

A HSC Mathematics. Students who have not completed HSC Mathematics (or equivalent) are strongly advised to take the Mathematics Bridging Course (offered in February). Please note: this unit does not normally lead to a major in Mathematics or Statistics or Financial Mathematics and Statistics. P NSW HSC 2 unit Mathematics or equivalent or a credit or above in MATH1111 N MATH1001 or MATH1901 or MATH1906 or BIOM1003 or ENVX1001 or MATH1021 or MATH1921 or MATH1931

Semester 1Summer Main

MATH1013Mathematical Modelling

3

A HSC Mathematics or a credit or higher in MATH1111. Students who have not completed HSC Mathematics (or equivalent) are strongly advised to take the Mathematics Bridging Course (offered in February). Please note: this unit does not normally lead to a major in Mathematics or Statistics or Financial Mathematics and Statistics. P NSW HSC 2 unit Mathematics or equivalent or a credit or above in MATH1111 N MATH1003 or MATH1903 or MATH1907 or MATH1023 or MATH1923 or MATH1933

Semester 2Summer Main

MATH1014Introduction to Linear Algebra

3

A HSC Mathematics or MATH1111. Students who have not completed HSC Mathematics (or equivalent) are strongly advised to take the Mathematics Bridging Course (offered in February). Please note: this unit does not normally lead to a major in Mathematics or Statistics or Financial Mathematics and Statistics. N MATH1012 or MATH1002 or MATH1902

Semester 2Summer Main

Regular level

MATH1021Calculus Of One Variable

3

A HSC Mathematics Extension 1 or equivalent. P NSW HSC 2 unit Mathematics or equivalent or a credit or above in MATH1111 N MATH1011 or MATH1901 or MATH1906 or ENVX1001 or MATH1001 or MATH1921 or MATH1931

Semester 1Semester 2Summer Main

MATH1002Linear Algebra

3

A HSC Mathematics or MATH1111. Students who have not completed HSC Mathematics (or equivalent) are strongly advised to take the Mathematics Bridging Course (offered in February). N MATH1012 or MATH1014 or MATH1902

Semester 1Summer Main

MATH1023Multivariable Calculus and Modelling

3

A MATH1X21, HSC Mathematics Extension 1 or equivalent. P NSW HSC 2 unit Mathematics or equivalent or a credit or above in MATH1111 N MATH1013 or MATH1903 or MATH1907 or MATH1003 or MATH1923 or MATH1933

Semester 1Semester 2Summer Main

MATH1004Discrete Mathematics

3

A HSC Mathematics or MATH1111. Students who have not completed HSC Mathematics (or equivalent) are strongly advised to take the Mathematics Bridging Course (offered in February). N MATH1904 or MATH1064 or MATH2011

Semester 2

MATH1005Statistical Thinking with Data

3

A HSC Mathematics. Students who have not completed HSC Mathematics (or equivalent) are strongly advised to take the Mathematics Bridging Course (offered in February). N MATH1015 or MATH1905 or STAT1021 or STAT1022 or ECMT1010 or ENVX1001 or ENVX1002 or BUSS1020 or DATA1001 or DATA1901

Semester 1Semester 2Summer Main

MATH1064Discrete Mathematics for Computation

6

N MATH1004 or MATH1904

Semester 2

MATH1001Differential Calculus This unit of study is not available in 2019

3

A HSC Mathematics Extension 1. Students who have not completed HSC Extension 1 Mathematics (or equivalent) are strongly advised to take the Extension 1 Mathematics Bridging Course (offered in February). C MATH1003 or MATH1903 N MATH1011 or MATH1901 or MATH1906 or MATH1111 or ENVX1001.

Semester 1Summer Main

MATH1003Integral Calculus and Modelling This unit of study is not available in 2019

3

A HSC Mathematics Extension 1 or MATH1001 or MATH1011 or a credit or higher in MATH1111. Students who have not completed HSC Extension 1 Mathematics (or equivalent) are strongly advised to take the Extension 1 Mathematics Bridging Course (offered in February). N MATH1013 or MATH1903 or MATH1907

Summer Main

Advanced level

MATH1921Calculus Of One Variable (Advanced)

3

A HSC Mathematics Extension 2 or equivalent. P NSW HSC 2 unit Mathematics or equivalent or a credit or above in MATH1111 N MATH1001 or MATH1011 or MATH1906 or ENVX1001 or MATH1901 or MATH1021 or MATH1931

Note: Department permission required for enrolment

Semester 1

MATH1902Linear Algebra (Advanced)

3

A (HSC Mathematics Extension 2) OR (90 or above in HSC Mathematics Extension 1) or equivalent N MATH1002 or MATH1012 or MATH1014

Note: Department permission required for enrolment

Semester 1

MATH1923Multivariable Calculus and Modelling (Adv)

3

A MATH1X21, HSC Mathematics Extension 2 or equivalent. P NSW HSC 2 unit Mathematics or equivalent or a credit or above in MATH1111 N MATH1003 or MATH1013 or MATH1907 or MATH1903 or MATH1023 or MATH1933

Note: Department permission required for enrolment

Semester 2

MATH1904Discrete Mathematics (Advanced)

3

A HSC Mathematics Extension 1. Students who have not completed HSC Extension 1 Mathematics (or equivalent) are strongly advised to take the Extension 1 Mathematics Bridging Course (offered in February). N MATH1004 or MATH1064 or MATH2011

Note: Department permission required for enrolment

Semester 2

MATH1905Statistical Thinking with Data (Advanced)

3

A (HSC Mathematics Extension 2) OR (90 or above in HSC Mathematics Extension 1) or equivalent N MATH1005 or MATH1015 or STAT1021 or STAT1022 or ECMT1010 or ENVX1001 or ENVX1002 or BUSS1020 or DATA1001 or DATA1901

Note: Department permission required for enrolment

Semester 2

Special Studies level

MATH1931Calculus Of One Variable (SSP)

3

A Band 4 in HSC Mathematics Extension 2 or equivalent. P NSW HSC 2 unit Mathematics or equivalent or a credit or above in MATH1111 N MATH1001 or MATH1011 or MATH1901 or ENVX1001 or MATH1906 or MATH1021 or MATH1921

Note: Department permission required for enrolmentEnrolment is by invitation only

Semester 1

MATH1933Multivariable Calculus and Modelling (SSP)

3

A MATH1X21, Band 4 in HSC Mathematics Extension 2 or equivalent. P NSW HSC 2 unit Mathematics or equivalent or a credit or above in MATH1111 N MATH1003 or MATH1903 or MATH1013 or MATH1907 or MATH1023 or MATH1923

Note: Department permission required for enrolmentEnrolment is by invitation only.

Semester 2

Intermediate units of study

MATH2021Vector Calculus and Differential Equations

6

P (MATH1X21 or MATH1931 or MATH1X01 or MATH1906) and (MATH1XX2) and (MATH1X23 or MATH1933 or MATH1X03 or MATH1907) N MATH2921 or MATH2065 or MATH2965 or (MATH2061 and MATH2022) or (MATH2061 and MATH2922) or (MATH2961 and MATH2022) or (MATH2961 and MATH2922) or MATH2067

Semester 1

MATH2921Vector Calculus and Differential Eqs (Adv)

6

P [(MATH1921 or MATH1931 or MATH1901 or MATH1906) or (a mark of 65 or above in MATH1021 or MATH1001)] and [MATH1902 or (a mark of 65 or above in MATH1002)] and [(MATH1923 or MATH1933 or MATH1903 or MATH1907) or (a mark of 65 or above in MATH1023 or MATH1003)] N MATH2021 or MATH2065 or MATH2965 or (MATH2061 and MATH2022) or (MATH2061 and MATH2922) or (MATH2961 and MATH2022) or (MATH2961 and MATH2922) or MATH2067

Semester 1

MATH2022Linear and Abstract Algebra

6

P MATH1XX2 N MATH2922 or MATH2968 or (MATH2061 and MATH2021) or (MATH2061 and MATH2921) or (MATH2961 and MATH2021) or (MATH2961 and MATH2921)

Semester 1

MATH2922Linear and Abstract Algebra (Advanced)

6

P MATH1902 or (a mark of 65 or above in MATH1002) N MATH2022 or MATH2968 or (MATH2061 and MATH2021) or (MATH2061 and MATH2921) or (MATH2961 and MATH2021) or (MATH2961 and MATH2921)

Semester 1

MATH2023Analysis

6

P (MATH1X21 or MATH1931 or MATH1X01 or MATH1906) and (MATH1X23 or MATH1933 or MATH1X03 or MATH1907) and (MATH1XX2) N MATH2923 or MATH3068 or MATH2962

Semester 2

MATH2923Analysis (Advanced)

6

P [(MATH1921 or MATH1931 or MATH1901 or MATH1906) or (a mark of 65 or above in MATH1021 or MATH1001)] and [MATH1902 or (a mark of 65 or above in MATH1002)] and [(MATH1923 or MATH1933 or MATH1903 or MATH1907) or (a mark of 65 or above in MATH1023 or MATH1003)] N MATH2023 or MATH2962 or MATH3068

Semester 2

MATH2061Linear Mathematics and Vector Calculus

6

P (MATH1X21 or MATH1011 or MATH1931 or MATH1X01 or MATH1906) and (MATH1014 or MATH1X02) and (MATH1X23 or MATH1933 or MATH1X03 or MATH1907) N MATH2001 or MATH2901 or MATH2002 or MATH2902 or MATH2961 or MATH2067 or MATH2021 or MATH2921 or MATH2022 or MATH2922

This unit of study is only available to Faculty of Engineering and Information Technologies students.

Semester 1Summer Main

MATH2088Number Theory and Cryptography

6

P (MATH1014 or MATH1902 or MATH1902) and (MATH1001 or MATH1901 or MATH1906 or MATH1003 or MATH1903 or MATH1907 or MATH1011 or MATH1021 or MATH1921 or MATH1931 or MATH1013 or MATH1023 or MATH1923 or MATH1933 or MATH1008 or MATH1904 or MATH1064) N MATH2068 or MATH2988

Semester 2

MATH2988Number Theory and Cryptography Adv

6

P [MATH19X1 or MATH1906 or (a mark of 65 or above in MATH1021 or MATH1001)] and [MATH19X3 or MATH1907 or (a mark of 65 or above in MATH1023 or MATH1003)] and [MATH1902 or (a mark of 65 or above in MATH1002)] N MATH2068 or MATH2088

Semester 2

MATH2070Optimisation and Financial Mathematics

6

A MATH1X23 or MATH1933 or MATH1X03 or MATH1907 P (MATH1X21 or MATH1011 or MATH1931 or MATH1X01 or MATH1906) and (MATH1014 or MATH1X02) N MATH2010 or MATH2033 or MATH2933 or MATH2970 or ECMT3510

Students may enrol in both MATH2070 and MATH3075 in the same semester

Semester 2

MATH2970Optimisation and Financial Mathematics Adv

6

A MATH19X3 or MATH1907 or a mark of 65 or above in MATH1003 or MATH1023 P [MATH19X1 or MATH1906 or (a mark of 65 or above in MATH1021 or MATH1001)] and [MATH1902 or (a mark of 65 or above in MATH1002)] N MATH2010 or MATH2033 or MATH2933 or MATH2070 or ECMT3510

Students may enrol in both MATH2970 and MATH3975 in the same semester

Semester 2

MATH2916Working Seminar A (SSP)

3

P High Distinction average over 12 credit points of Junior Advanced Mathematics

Note: Department permission required for enrolment

Semester 1

MATH2917Working Seminar B (SSP)

3

P High Distinction average over 12 credit points of Junior Advanced Mathematics

Note: Department permission required for enrolment

Semester 2

Senior units of study

MATH3061Geometry and Topology

6

P 12 credit points of Intermediate Mathematics N MATH3001 or MATH3006

Semester 2

MATH3063Nonlinear ODEs with Applications

6

A MATH2061 or MATH2961 or [MATH2X21 and MATH2X22] P 12 credit points of MATH2XXX units of study N MATH3963 or MATH4063

Semester 1

MATH3963Nonlinear ODEs with Applications (Adv)

6

A MATH2061 or MATH2961 or (MATH2X21 and MATH2X22) P A mark of 65 or greater in 12 credit points of MATH2XXX units of study N MATH3063 or MATH4063

A MATH2X21 and MATH2X23 and STAT2X11 P [A mark of 65 or greater in 12cp from (MATH2070 or MATH2970 or STAT2011 or STAT2911 or MATH2021 or MATH2921 or MATH2022 or MATH2922 or MATH2023 or MATH2923 or MATH2061 or MATH2961 or MATH2065 or MATH2965 or MATH2962 or STAT2012 or STAT2912 or DATA2002 or DATA2902)] or [12 cp from (MATH3075 or MATH3975 or STAT3021 or STAT3011 or STAT3911 or STAT3888 or STAT3014 or STAT3914 or MATH3063 or MATH3963 or MATH3061 or MATH3961 or MATH3962 or MATH3963 or MATH3969 or MATH3974 or MATH3976 or MATH3977 or MATH3978 or MATH3979)] N MATH4071

P Credit average or greater in 12 credit points of Intermediate Mathematics (including MATH2070 or MATH2970) N MATH3933 or MATH3015 or MATH3075

Semester 2

MATH3076Mathematical Computing

6

P 12 credit points of MATH2XXX and 6 credit points from (MATH1021 or MATH1001 or MATH1023 or MATH1003 or MATH19X1 or MATH19X3 or MATH1906 or MATH1907) N MATH3976 or MATH4076

Semester 1

MATH3976Mathematical Computing (Advanced)

6

P 12 credit points of MATH2XXX and [3 credit points from (MATH1923 or MATH1903 or MATH1933 or MATH1907), or a mark of 65 or above in (MATH1023 or MATH1003)] N MATH3076 or MATH4076

Semester 1

MATH3078PDEs and Waves

6

A [MATH2X61 and MATH2X65] or [MATH2X21 and MATH2X22] P 12 credit points of MATH2XXX units of study N MATH3978 or MATH4078

Semester 2

MATH3978PDEs and Waves (Advanced)

6

A [MATH2X61 and MATH2X65] or [MATH2X21 and MATH2X22] P A mark of 65 or greater in 12 credit points of MATH2XXX units of study N MATH3078 or MATH4078

Semester 2

MATH3961Metric Spaces (Advanced)

6

A MATH2923 or MATH2962 or MATH2023 or MATH3068 P A mark of 65 or greater in 12 credit points of 2000-level Mathematics units N MATH4061

Semester 1

MATH3962Rings, Fields and Galois Theory (Adv)

6

A MATH2922 or MATH2961 P MATH2961 or MATH2922 or a a mark of 65 or greater in (MATH2061 or MATH2022) N MATH3062 or MATH4062

Students are advised to take MATH2968 before attempting this unit.

Semester 1

MATH3968Differential Geometry (Advanced)

6

A (MATH2921 and MATH2922) or MATH2961 P A mark of 65 or greater in 12 credit points of MATH2XXX units of study N MATH4068

Semester 2

MATH3969Measure Theory and Fourier Analysis (Adv)

6

A (MATH2921 and MATH2922) or MATH2961 P A mark of 65 or greater in 12 credit points of MATH2XXX units of study N MATH4069

Semester 2

MATH3974Fluid Dynamics (Advanced)

6

A [MATH2961 and MATH2965] or [MATH2921 and MATH2922] P Credit average or greater in 12 credit points of Intermediate Mathematics N MATH4074

Semester 1

MATH3977Lagrangian and Hamiltonian Dynamics (Adv)

6

P A mark of 65 or greater in 12 credit points of MATH2XXX units of study N MATH4077

Semester 2

MATH3979Complex Analysis (Advanced)

6

A MATH2023 or MATH2923 or MATH2962 or MATH3068 P [A mark of 65 or greater in 12cp from (MATH2021 or MATH2921 or MATH2022 or MATH2922 or MATH2023 or MATH2923 or MATH2061 or MATH2961 or MATH2065 or MATH2965 or MATH2962)] N MATH4079 or MATH3964

Semester 1

MATH3888Projects in Mathematics

6

P (MATH2921 or MATH2021 or MATH2065 or MATH2965 or MATH2061 or MATH2961 or MATH2923 or MATH2023) and (MATH2922 or MATH2022 or MATH2061 or MATH2961 or MATH2088 or MATH2988)

Note: Students who have previously successfully studied calculus at a level at least equivalent to HSC Mathematics are prohibited.

This unit is an introduction to the calculus of one variable. Topics covered include elementary functions, differentiation, basic integration techniques and coordinate geometry in three dimensions. Applications in science and engineering are emphasised.

Textbooks

As set out in the Junior Mathematics Handbook

Fundamental level

MATH1011 Applications of Calculus

Credit points: 3 Teacher/Coordinator: A/Prof Sharon Stephen Session: Semester 1,Summer Main Classes: 2x1-hr lectures; 1x1-hr tutorial/wk Prerequisites: NSW HSC 2 unit Mathematics or equivalent or a credit or above in MATH1111 Prohibitions: MATH1001 or MATH1901 or MATH1906 or BIOM1003 or ENVX1001 or MATH1021 or MATH1921 or MATH1931 Assumed knowledge: HSC Mathematics. Students who have not completed HSC Mathematics (or equivalent) are strongly advised to take the Mathematics Bridging Course (offered in February). Please note: this unit does not normally lead to a major in Mathematics or Statistics or Financial Mathematics and Statistics. Assessment: 2 x quizzes (30%); 2 x assignments (5%); final exam (65%) Campus: Camperdown/Darlington, Sydney Mode of delivery: Normal (lecture/lab/tutorial) day

This unit is designed for science students who do not intend to undertake higher year mathematics and statistics. It establishes and reinforces the fundamentals of calculus, illustrated where possible with context and applications. Specifically, it demonstrates the use of (differential) calculus in solving optimisation problems and of (integral) calculus in measuring how a system accumulates over time. Topics studied include the fitting of data to various functions, the interpretation and manipulation of periodic functions and the evaluation of commonly occurring summations. Differential calculus is extended to functions of two variables and integration techniques include integration by substitution and the evaluation of integrals of infinite type.

Textbooks

Applications of Calculus (Course Notes for MATH1011)

MATH1013 Mathematical Modelling

Credit points: 3 Teacher/Coordinator: A/Prof Sharon Stephen Session: Semester 2,Summer Main Classes: 2x1-hr lectures; 1x1-hr tutorial/wk Prerequisites: NSW HSC 2 unit Mathematics or equivalent or a credit or above in MATH1111 Prohibitions: MATH1003 or MATH1903 or MATH1907 or MATH1023 or MATH1923 or MATH1933 Assumed knowledge: HSC Mathematics or a credit or higher in MATH1111. Students who have not completed HSC Mathematics (or equivalent) are strongly advised to take the Mathematics Bridging Course (offered in February). Please note: this unit does not normally lead to a major in Mathematics or Statistics or Financial Mathematics and Statistics. Assessment: 2 x quizzes (30%); 2 x assignments (5%); online homework (10%); final exam (55%) Campus: Camperdown/Darlington, Sydney Mode of delivery: Normal (lecture/lab/tutorial) day

MATH1013 is designed for science students who do not intend to undertake higher year mathematics and statistics.In this unit of study students learn how to construct, interpret and solve simple differential equations and recurrence relations. Specific techniques include separation of variables, partial fractions and first and second order linear equations with constant coefficients. Students are also shown how to iteratively improve approximate numerical solutions to equations.

Textbooks

Mathematical Modelling, Leon Poladian, School of Mathematics and Statistics, University of Sydney (2011)

MATH1014 Introduction to Linear Algebra

Credit points: 3 Teacher/Coordinator: A/Prof Sharon Stephen Session: Semester 2,Summer Main Classes: 2x1-hr lectures; 1x1-hr tutorial/wk Prohibitions: MATH1012 or MATH1002 or MATH1902 Assumed knowledge: HSC Mathematics or MATH1111. Students who have not completed HSC Mathematics (or equivalent) are strongly advised to take the Mathematics Bridging Course (offered in February). Please note: this unit does not normally lead to a major in Mathematics or Statistics or Financial Mathematics and Statistics. Assessment: 2 x quizzes (30%); 2 x assignments (5%); final exam (65%) Campus: Camperdown/Darlington, Sydney Mode of delivery: Normal (lecture/lab/tutorial) day

This unit is an introduction to Linear Algebra. Topics covered include vectors, systems of linear equations, matrices, eigenvalues and eigenvectors. Applications in life and technological sciences are emphasised.

Textbooks

A First Course in Linear Algebra (3rd edition), David Easdown, Pearson Education (2010)

Calculus is a discipline of mathematics that finds profound applications in science, engineering, and economics. This unit investigates differential calculus and integral calculus of one variable and the diverse applications of this theory. Emphasis is given both to the theoretical and foundational aspects of the subject, as well as developing the valuable skill of applying the mathematical theory to solve practical problems. Topics covered in this unit of study include complex numbers, functions of a single variable, limits and continuity, differentiation, optimisation, Taylor polynomials, Taylor's Theorem, Taylor series, Riemann sums, and Riemann integrals.

MATH1002 is designed to provide a thorough preparation for further study in mathematics and statistics. It is a core unit of study providing three of the twelve credit points required by the Faculty of Science as well as a Junior level requirement in the Faculty of Engineering.This unit of study introduces vectors and vector algebra, linear algebra including solutions of linear systems, matrices, determinants, eigenvalues and eigenvectors.

Calculus is a discipline of mathematics that finds profound applications in science, engineering, and economics. This unit investigates multivariable differential calculus and modelling. Emphasis is given both to the theoretical and foundational aspects of the subject, as well as developing the valuable skill of applying the mathematical theory to solve practical problems. Topics covered in this unit of study include mathematical modelling, first order differential equations, second order differential equations, systems of linear equations, visualisation in 2 and 3 dimensions, partial derivatives, directional derivatives, the gradient vector, and optimisation for functions of more than one variable.

MATH1004 is designed to provide a thorough preparation for further study in Mathematics. This unit provides an introduction to fundamental aspects of discrete mathematics, which deals with 'things that come in chunks that can be counted'. It focuses on the enumeration of a set of numbers, viz. Catalan numbers. Topics include sets and functions, counting principles, discrete probability, Boolean expressions, mathematical induction, linear recurrence relations, graphs and trees.

In a data-rich world, global citizens need to problem solve with data, and evidence based decision-making is essential is every field of research and work. This unit equips you with the foundational statistical thinking to become a critical consumer of data. You will learn to think analytically about data and to evaluate the validity and accuracy of any conclusions drawn. Focusing on statistical literacy, the unit covers foundational statistical concepts, including the design of experiments, exploratory data analysis, sampling and tests of significance.

This unit introduces students to the language and key methods of the area of Discrete Mathematics. The focus is on mathematical concepts in discrete mathematics and their applications, with an emphasis on computation. For instance, to specify a computational problem precisely one needs to give an abstract formulation using mathematical objects such as sets, functions, relations, orders, and sequences. In order to prove that a proposed solution is correct, one needs to apply the principles of mathematical logic, and to use proof techniques such as induction. To reason about the efficiency of an algorithm, one often needs to estimate the growth of functions or count the size of complex mathematical objects. This unit provides the necessary mathematical background for such applications of discrete mathematics. Students will be introduced to mathematical logic and proof techniques; sets, functions, relations, orders, and sequences; counting and discrete probability; asymptotic growth; and basic graph theory.

MATH1001 is designed to provide a thorough preparation for further study in mathematics and statistics. It is a core unit of study providing three of the twelve credit points required by the Faculty of Science as well as a Junior level requirement in the Faculty of Engineering. This unit of study looks at complex numbers, functions of a single variable, limits and continuity, vector functions and functions of two variables. Differential calculus is extended to functions of two variables. Taylor's theorem as a higher order mean value theorem.

Textbooks

As set out in the Junior Mathematics Handbook.

MATH1003 Integral Calculus and Modelling

This unit of study is not available in 2019

Credit points: 3 Session: Summer Main Classes: Two 1 hour lectures and one 1 hour tutorial per week. Prohibitions: MATH1013 or MATH1903 or MATH1907 Assumed knowledge: HSC Mathematics Extension 1 or MATH1001 or MATH1011 or a credit or higher in MATH1111. Students who have not completed HSC Extension 1 Mathematics (or equivalent) are strongly advised to take the Extension 1 Mathematics Bridging Course (offered in February). Assessment: One 1.5 hour examination, assignments and quizzes (100%) Campus: Camperdown/Darlington, Sydney Mode of delivery: Normal (lecture/lab/tutorial) day

MATH1003 is designed to provide a thorough preparation for further study in mathematics and statistics. It is a core unit of study providing three of the twelve credit points required by the Faculty of Science as well as a Junior level requirement in the Faculty of Engineering.This unit of study first develops the idea of the definite integral from Riemann sums, leading to the Fundamental Theorem of Calculus. Various techniques of integration are considered, such as integration by parts.The second part is an introduction to the use of first and second order differential equations to model a variety of scientific phenomena.

Calculus is a discipline of mathematics that finds profound applications in science, engineering, and economics. This unit investigates differential calculus and integral calculus of one variable and the diverse applications of this theory. Emphasis is given both to the theoretical and foundational aspects of the subject, as well as developing the valuable skill of applying the mathematical theory to solve practical problems. Topics covered in this unit of study include complex numbers, functions of a single variable, limits and continuity, differentiation, optimisation, Taylor polynomials, Taylor's Theorem, Taylor series, Riemann sums, and Riemann integrals. Additional theoretical topics included in this advanced unit include the Intermediate Value Theorem, Rolle's Theorem, and the Mean Value Theorem.

This unit is designed to provide a thorough preparation for further study in mathematics and statistics. It is a core unit of study providing three of the twelve credit points required by the Faculty of Science as well as a Junior level requirement in the Faculty of Engineering. It parallels the normal unit MATH1002 but goes more deeply into the subject matter and requires more mathematical sophistication.

Calculus is a discipline of mathematics that finds profound applications in science, engineering, and economics. This unit investigates multivariable differential calculus and modelling. Emphasis is given both to the theoretical and foundational aspects of the subject, as well as developing the valuable skill of applying the mathematical theory to solve practical problems. Topics covered in this unit of study include mathematical modelling, first order differential equations, second order differential equations, systems of linear equations, visualisation in 2 and 3 dimensions, partial derivatives, directional derivatives, the gradient vector, and optimisation for functions of more than one variable. Additional topics covered in this advanced unit of study include the use of diagonalisation of matrices to study systems of linear equation and optimisation problems, limits of functions of two or more variables, and the derivative of a function of two or more variables.

This unit is designed to provide a thorough preparation for further study in mathematics. It parallels the normal unit MATH1004 but goes more deeply into the subject matter and requires more mathematical sophistication.

This unit is designed to provide a thorough preparation for further study in mathematics and statistics. It is a core unit of study providing three of the twelve credit points required by the Faculty of Science as well as a Junior level requirement in the Faculty of Engineering. This Advanced level unit of study parallels the normal unit MATH1005 but goes more deeply into the subject matter and requires more mathematical sophistication.

The Mathematics Special Studies Program is for students with exceptional mathematical aptitude, and requires outstanding performance in past mathematical studies. Students will cover the material of MATH1921 Calculus of One Variable (Adv), and attend a weekly seminar covering special topics on available elsewhere in the Mathematics and Statistics program.

The Mathematics Special Studies Program is for students with exceptional mathematical aptitude, and requires outstanding performance in past mathematical studies. Students will cover the material of MATH1923 Multivariable Calculus and Modelling (Adv), and attend a weekly seminar covering special topics on available elsewhere in the Mathematics and Statistics program.

This unit opens with topics from vector calculus, including vector-valued functions (parametrised curves and surfaces; vector fields; div, grad and curl; gradient fields and potential functions), line integrals (arc length; work; path-independent integrals and conservative fields; flux across a curve), iterated integrals (double and triple integrals, polar, cylindrical and spherical coordinates; areas, volumes and mass; Green's Theorem), flux integrals (flow through a surface; flux integrals through a surface defined by a function of two variables, through cylinders, spheres and other parametrised surfaces), Gauss' and Stokes' theorems. The unit then moves to topics in solution techniques for ordinary and partial differential equations (ODEs and PDEs) with applications. It provides a basic grounding in these techniques to enable students to build on the concepts in their subsequent courses. The main topics are: second order ODEs (including inhomogeneous equations), higher order ODEs and systems of first order equations, solution methods (variation of parameters, undetermined coefficients) the Laplace and Fourier Transform, an introduction to PDEs, and first methods of solutions (including separation of variables, and Fourier Series).

Textbooks

As set out in the Intermediate Mathematics Handbook

MATH2921 Vector Calculus and Differential Eqs (Adv)

Credit points: 6 Teacher/Coordinator: Dr Daniel Hauer Session: Semester 1 Classes: 3x1-hr lectures; 1x1-hr tutorial; and 1x1-hr practice class/wk Prerequisites: [(MATH1921 or MATH1931 or MATH1901 or MATH1906) or (a mark of 65 or above in MATH1021 or MATH1001)] and [MATH1902 or (a mark of 65 or above in MATH1002)] and [(MATH1923 or MATH1933 or MATH1903 or MATH1907) or (a mark of 65 or above in MATH1023 or MATH1003)] Prohibitions: MATH2021 or MATH2065 or MATH2965 or (MATH2061 and MATH2022) or (MATH2061 and MATH2922) or (MATH2961 and MATH2022) or (MATH2961 and MATH2922) or MATH2067 Assessment: Quizzes (10%), assignments (20%); final exam (60%) Campus: Camperdown/Darlington, Sydney Mode of delivery: Normal (lecture/lab/tutorial) day

This is the advanced version of MATH2021, with more emphasis on the underlying concepts and mathematical rigour. The vector calculus component of the course will include: parametrised curves and surfaces, vector fields, div, grad and curl, gradient fields and potential functions, lagrange multipliers line integrals, arc length, work, path-independent integrals, and conservative fields, flux across a curve, double and triple integrals, change of variable formulas, polar, cylindrical and spherical coordinates, areas, volumes and mass, flux integrals, and Green's Gauss' and Stokes' theorems. The Differential Equations half of the course will focus on ordinary and partial differential equations (ODEs and PDEs) with applications with more complexity and depth. The main topics are: second order ODEs (including inhomogeneous equations), series solutions near a regular point, higher order ODEs and systems of first order equations, matrix equations and solutions, solution methods (variation of parameters, undetermined coefficients) the Laplace and Fourier Transform, elementary Sturm-Liouville theory, an introduction to PDEs, and first methods of solutions (including separation of variables, and Fourier Series). The unit then moves to topics in solution techniques for ordinary and partial differential equations (ODEs and PDEs) with applications. It provides a more thorough grounding in these techniques to enable students to build on the concepts in their subsequent courses. The main topics are: second order ODEs (including inhomogeneous equations), higher order ODEs and systems of first order equations, solution methods (variation of parameters, undetermined coefficients) the Laplace and Fourier Transform, an introduction to PDEs, and first methods of solutions (including separation of variables, and Fourier Series).

Linear and abstract algebra is one of the cornerstones of mathematics and it is at the heart of many applications of mathematics and statistics in the sciences and engineering. This unit investigates and explores properties of linear functions, developing general principles relating to the solution sets of homogeneous and inhomogeneous linear equations, including differential equations. Linear independence is introduced as a way of understanding and solving linear systems of arbitrary dimension. Linear operators on real spaces are investigated, paying particular attention to the geometrical significance of eigenvalues and eigenvectors, extending ideas from first year linear algebra. To better understand symmetry, matrix and permutation groups are introduced and used to motivate the study of abstract group theory.

Linear and abstract algebra is one of the cornerstones of mathematics and it is at the heart of many applications of mathematics and statistics in the sciences and engineering. This unit is an advanced version of MATH2022, with more emphasis on the underlying concepts and on mathematical rigour. This unit investigates and explores properties of vector spaces, matrices and linear transformations, developing general principles relating to the solution sets of homogeneous and inhomogeneous linear equations, including differential equations. Linear independence is introduced as a way of understanding and solving linear systems of arbitrary dimension. Linear operators on real spaces are investigated, paying particular attention to the geometrical significance of eigenvalues and eigenvectors, extending ideas from first year linear algebra. To better understand symmetry, matrix and permutation groups are introduced and used to motivate the study of abstract groups theory. The unit culminates in studying inner spaces, quadratic forms and normal forms of matrices together with their applications to problems both in mathematics and in the sciences and engineering.

Analysis grew out of calculus, which leads to the study of limits of functions, sequences and series. It is one of the fundamental topics underlying much of mathematics including differential equations, dynamical systems, differential geometry, topology and Fourier analysis. This unit introduces the field of mathematical analysis both with a careful theoretical framework as well as selected applications. It shows the utility of abstract concepts and teaches an understanding and construction of proofs in mathematics. This unit will be useful to students of mathematics, science and engineering and in particular to future school mathematics teachers, because we shall explain why common practices in the use of calculus are correct, and understanding this is important for correct applications and explanations. The unit starts with the foundations of calculus and the real numbers system. It goes on to study the limiting behaviour of sequences and series of real and complex numbers. This leads naturally to the study of functions defined as limits and to the notion of uniform convergence. Returning to the beginnings of calculus and power series expansions leads to complex variable theory: elementary functions of complex variable, the Cauchy integral theorem, Cauchy integral formula, residues and related topics with applications to real integrals.

Textbooks

As set out in the Intermediate Mathematics Handbook

MATH2923 Analysis (Advanced)

Credit points: 6 Teacher/Coordinator: Dr Daniel Hauer Session: Semester 2 Classes: 3x1-hr lectures; 1x1-hr practice class and 1x1-hr tutorial/wk Prerequisites: [(MATH1921 or MATH1931 or MATH1901 or MATH1906) or (a mark of 65 or above in MATH1021 or MATH1001)] and [MATH1902 or (a mark of 65 or above in MATH1002)] and [(MATH1923 or MATH1933 or MATH1903 or MATH1907) or (a mark of 65 or above in MATH1023 or MATH1003)] Prohibitions: MATH2023 or MATH2962 or MATH3068 Assessment: 2 x quizzes (30%); an assignment (10%); and a final exam (60%) Campus: Camperdown/Darlington, Sydney Mode of delivery: Normal (lecture/lab/tutorial) day

Analysis grew out of calculus, which leads to the study of limits of functions, sequences and series. It is one of the fundamental topics underlying much of mathematics including differential equations, dynamical systems, differential geometry, topology and Fourier analysis. This advanced unit introduces the field of mathematical analysis both with a careful theoretical frame- work as well as selected applications.This unit will be useful to students with more mathematical maturity who study mathematics, science, or engineering. Starting off with an axiomatic description of the real numbers system, this unit concentrates on the limiting behaviour of sequences and series of real and complex numbers. This leads naturally to the study of functions defined as limits and to the notion of uniform con-vergence. Special attention is given to power series, leading into the theory of analytic functions and complex analysis. Besides a rigourous treatment of many concepts from calculus, you will learn the basic results of complex analysis such as the Cauchy integral theorem, Cauchy integral formula, the residues theorems, leading to useful techniques for evaluating real integrals. By doing this unit, you will develop solid foundations in the more formal aspects of analysis, including knowledge of abstract concepts, how to apply them and the ability to construct proofs in mathematics.

Note: This unit of study is only available to Faculty of Engineering and Information Technologies students.

This unit starts with an investigation of linearity: linear functions, general principles relating to the solution sets of homogeneous and inhomogeneous linear equations (including differential equations), linear independence and the dimension of a linear space. The study of eigenvalues and eigenvectors, begun in junior level linear algebra, is extended and developed. The unit then moves on to topics from vector calculus, including vector-valued functions (parametrised curves and surfaces; vector fields; div, grad and curl; gradient fields and potential functions), line integrals (arc length; work; path-independent integrals and conservative fields; flux across a curve), iterated integrals (double and triple integrals; polar, cylindrical and spherical coordinates; areas, volumes and mass; Green's Theorem), flux integrals (flow through a surface; flux integrals through a surface defined by a function of two variables, though cylinders, spheres and parametrised surfaces), Gauss' Divergence Theorem and Stokes' Theorem.

Textbooks

Course Notes for MATH2061 Vector Calculus, S Britton and K-G Choo

MATH2088 Number Theory and Cryptography

Credit points: 6 Teacher/Coordinator: Dr Daniel Hauer Session: Semester 2 Classes: Three 1 hour lectures, one 1 hour tutorial and one 1 hour computer laboratory per week. Prerequisites: (MATH1014 or MATH1902 or MATH1902) and (MATH1001 or MATH1901 or MATH1906 or MATH1003 or MATH1903 or MATH1907 or MATH1011 or MATH1021 or MATH1921 or MATH1931 or MATH1013 or MATH1023 or MATH1923 or MATH1933 or MATH1008 or MATH1904 or MATH1064) Prohibitions: MATH2068 or MATH2988 Assessment: 2 hour exam, assignments, quizzes (100%) Campus: Camperdown/Darlington, Sydney Mode of delivery: Normal (lecture/lab/tutorial) day

Cryptography is the branch of mathematics that provides the techniques for confidential exchange of information sent via possibly insecure channels. This unit introduces the tools from elementary number theory that are needed to understand the mathematics underlying the most commonly used modern public key cryptosystems. Topics include the Euclidean Algorithm, Fermat's Little Theorem, the Chinese Remainder Theorem, Mobius Inversion, the RSA Cryptosystem, the Elgamal Cryptosystem and the Diffie-Hellman Protocol. Issues of computational complexity are also discussed.

Note: Students may enrol in both MATH2070 and MATH3075 in the same semester

Problems in industry and commerce often involve maximising profits or minimising costs subject to constraints arising from resource limitations. The first part of this unit looks at programming problems and their solution using the simplex algorithm; nonlinear optimisation and the Kuhn Tucker conditions.The second part of the unit deals with utility theory and modern portfolio theory. Topics covered include: pricing under the principles of expected return and expected utility; mean-variance Markowitz portfolio theory, the Capital Asset Pricing Model, log-optimal portfolios and the Kelly criterion; dynamical programming. Some understanding of probability theory including distributions and expectations is required in this part.Theory developed in lectures will be complemented by computer laboratory sessions using MATLAB. Minimal computing experience will be required.

MATH2970 Optimisation and Financial Mathematics Adv

Credit points: 6 Teacher/Coordinator: Prof Martin Wechslberger Session: Semester 2 Classes: 3x1-hr lectures; 1x1-hr tutorial; and 1x1-hr computer lab/wk (lectures given in common with MATH2070). Prerequisites: [MATH19X1 or MATH1906 or (a mark of 65 or above in MATH1021 or MATH1001)] and [MATH1902 or (a mark of 65 or above in MATH1002)] Prohibitions: MATH2010 or MATH2033 or MATH2933 or MATH2070 or ECMT3510 Assumed knowledge: MATH19X3 or MATH1907 or a mark of 65 or above in MATH1003 or MATH1023 Assessment: 1 x2-hr exam (70%), 1 x assignment (10%), 1 x quiz (10%); 1 x computational project (10%). To pass the course at least 50% in the final exam is necessary. Campus: Camperdown/Darlington, Sydney Mode of delivery: Normal (lecture/lab/tutorial) day

Note: Students may enrol in both MATH2970 and MATH3975 in the same semester

The content of this unit of study parallels that of MATH2070, but students enrolled at Advanced level will undertake more advanced problem solving and assessment tasks, and some additional topics may be included.

The main aim of this unit is to develop the students' written and oral presentation skills. The material will consist of a series of connected topics relevant to modern mathematics and statistics. The topics are chosen to suit the students' background and interests, and are not covered by other mathematics or statistics units. The first session will be an introduction on the principles of written and oral presentation of mathematics. Under the supervision and advice of the lecturer(s) in charge, the students present the topics to the other students and the lecturer in a seminar series and a written essay in a manner that reflects the practice of research in mathematics and statistics.

The main aim of this unit is to develop the students' written and oral presentation skills. The material will consist of a series of connected topics relevant to modern mathematics and statistics. The topics are chosen to suit the students' background and interests, and are not covered by other mathematics or statistics units. The first session will be an introduction on the principles of written and oral presentation of mathematics. Under the supervision and advice of the lecturer(s) in charge, the students present the topics to the other students and the lecturer in a seminar series and a written essay in a manner that reflects the practice of research in mathematics and statistics.

The aim of the unit is to expand visual/geometric ways of thinking. The Geometry section is concerned mainly with transformations of the Euclidean plane (that is, bijections from the plane to itself), with a focus on the study of isometries (proving the classification theorem for transformations which preserve distances between points), symmetries (including the classification of frieze groups) and affine transformations (transformations which map lines to lines). The basic approach is via vectors and matrices, emphasising the interplay between geometry and linear algebra. The study of affine transformations is then extended to the study of collineations in the real projective plane, including collineations which map conics to conics. The Topology section considers graphs, surfaces and knots from a combinatorial point of view. Key ideas such as homeomorphism, subdivision, cutting and pasting and the Euler invariant are introduced first for graphs (1-dimensional objects) and then for triangulated surfaces (2-dimensional objects). Topics include the classification of surfaces, map colouring, decomposition of knots and knot invariants.

This unit of study is an introduction to the theory of systems of ordinary differential equations. Such systems model many types of phenomena in engineering, biology and the physical sciences. The emphasis will not be on finding explicit solutions, but instead on the qualitative features of these systems, such as stability, instability and oscillatory behaviour. The aim is to develop a good geometrical intuition into the behaviour of solutions to such systems. Some background in linear algebra, and familiarity with concepts such as limits and continuity, will be assumed. The applications in this unit will be drawn from predator-prey systems, transmission of diseases, chemical reactions and other equations and systems from mathematical biology.

The theory of ordinary differential equations is a classical topic going back to Newton and Leibniz. It comprises a vast number of ideas and methods of different nature. The theory has many applications and stimulates new developments in almost all areas of mathematics. The emphasis is on qualitative analysis including phase-plane methods, bifurcation theory and the study of limit cycles. The more theoretical part includes existence and uniqueness theorems, linearisation, and analysis of asymptotic behaviour. The applications in this unit will be drawn from predator-prey systems, population models, chemical reactions, and other equations and systems from mathematical biology.

This unit of study unifies and extends mathematical ideas and techniques that most participants will have met in their first and second years, and will be of general interest to all students of pure and applied mathematics. It combines algebra and logic to present and answer a number of related questions of fundamental importance in the development of mathematics, from ancient to modern times.The Propositional and Predicate Calculi are studied as model axiomatic systems in their own right, including proofs of consistency and completeness. The final part of the course introduces precise notions of computability and decidability, through abstract Turing machines, culminating in the unsolvability of the Halting Problem the undecidability of First Order Logic, and a discussion of Godel's Incompleteness Theorem.Classical and novel arithmetics are introduced, unified and described abstractly using field and ring axioms and the language of field extensions. Quotient rings are introduced, which are used to construct different finite and infinite fields. A construction of the real numbers, by factoring out rings of Cauchy sequences of rationals by the ideal of null sequences, is presented. Axiomatics are placed in the context of reasoning within first order logic and set theory.

Analysis grew out of calculus, which leads to the study of limits of functions, sequences and series. The aim of the unit is to present enduring beautiful and practical results that continue to justify and inspire the study of analysis. The unit starts with the foundations of calculus and the real number system. It goes on to study the limiting behaviour of sequences and series of real and complex numbers. This leads naturally to the study of functions defined as limits and to the notion of uniform convergence. Returning to the beginnings of calculus and power series expansions leads to complex variable theory: analytic functions, Taylor expansions and the Cauchy Integral Theorem.Power series are not adequate to solve the problem of representing periodic phenomena such as wave motion. This requires Fourier theory, the expansion of functions as sums of sines and cosines. This unit deals with this theory, Parseval's identity, pointwise convergence theorems and applications.The unit goes on to introduce Bernoulli numbers, Bernoulli polynomials, the Euler MacLaurin formula and applications, the gamma function and the Riemann zeta function. Lastly we return to the foundations of analysis, and study limits from the point of view of topology.

MATH3971 Convex Analysis and Optimal Control (Adv)

Credit points: 6 Teacher/Coordinator: Prof Georg Gottwald Session: Semester 1 Classes: Lecture 3hours/week, tutorial 1hr/week Prerequisites: [A mark of 65 or greater in 12cp from (MATH2070 or MATH2970 or STAT2011 or STAT2911 or MATH2021 or MATH2921 or MATH2022 or MATH2922 or MATH2023 or MATH2923 or MATH2061 or MATH2961 or MATH2065 or MATH2965 or MATH2962 or STAT2012 or STAT2912 or DATA2002 or DATA2902)] or [12 cp from (MATH3075 or MATH3975 or STAT3021 or STAT3011 or STAT3911 or STAT3888 or STAT3014 or STAT3914 or MATH3063 or MATH3963 or MATH3061 or MATH3961 or MATH3962 or MATH3963 or MATH3969 or MATH3974 or MATH3976 or MATH3977 or MATH3978 or MATH3979)] Prohibitions: MATH4071 Assumed knowledge: MATH2X21 and MATH2X23 and STAT2X11 Assessment: Assignment (15%), assignment (15%), exam (70%) Campus: Camperdown/Darlington, Sydney Mode of delivery: Normal (lecture/lab/tutorial) day

The questions how to maximise your gain (or to minimise the cost) and how to determine the optimal strategy/policy are fundamental for an engineer, an economist, a doctor designing a cancer therapy, or a government planning some social policies. Many problems in mechanics, physics, neuroscience and biology can be formulated as optimistion problems. Therefore, optimisation theory is an indispensable tool for an applied mathematician. Optimisation theory has many diverse applications and requires a wide range of tools but there are only a few ideas underpinning all this diversity of methods and applications. This course will focus on two of them. We will learn how the concept of convexity and the concept of dynamic programming provide a unified approach to a large number of seemingly unrelated problems. By completing this unit you will learn how to formulate optimisation problems that arise in science, economics and engineering and to use the concepts of convexity and the dynamic programming principle to solve straight forward examples of such problems. You will also learn about important classes of optimisation problems arising in finance, economics, engineering and insurance.

This unit will introduce you to the mathematical theory of modern finance with the special emphasis on the valuation and hedging of financial derivatives, such as: forward contracts and options of European and American style. You will learn about the concept of arbitrage and how to model risk-free and risky securities. Topics covered by this unit include: notions of a martingale and a martingale measure, the fundamental theorems of asset pricing, complete and incomplete markets, the binomial options pricing model, discrete random walks and the Brownian motion, the Black-Scholes options pricing model and the valuation and heding of exotic options. Students completing this unit have been highly sought by the finance industry, which continues to need graduates with quantitative skills. Lectures in the mainstream unit are held concurrently with those of the corresponding advanced unit.

This unit will introduce you to the mathematical theory of modern finance with the special emphasis on the valuation and hedging of financial derivatives, such as: forward contracts and options of European and American style. You will learn about the concept of arbitrage and how to model risk-free and risky securities. Topics covered by this unit include: the notions of a martingale and a martingale measure, the fundamental theorems of asset pricing, complete and incomplete markets, the binomial options pricing model, discrete random walks and the Brownian motion, the Black-Scholes options pricing model and the valuation and heding of exotic options. Students completing this unit have been highly sought by the finance industry, which continues to need graduates with quantitative skills. Students enrolled in this unit at advanced level will have to undertake more challenging assessment tasks, but lectures in the advanced level are held concurrently with those of the corresponding mainstream unit.

This unit of study introduces Sturm-Liouville eigenvalue problems and their role in finding solutions to boundary value problems. Analytical solutions of linear PDEs are found using separation of variables and integral transform methods. Three of the most important equations of mathematical physics - the wave equation, the diffusion (heat) equation and Laplace's equation - are treated, together with a range of applications. There is particular emphasis on wave phenomena, with an introduction to the theory of nonlinear waves, dimensional analysis, symmetry.

This unit of study introduces Sturm-Liouville eigenvalue problems and their role in finding solutions to boundary value problems. Analytical solutions of linear PDEs are found using separation of variables and integral transform methods. Three of the most important equations of mathematical physics - the wave equation, the diffusion (heat) equation and Laplace's equation - are treated, together with a range of applications. There is particular emphasis on wave phenomena, with an introduction to the theory of nonlinear waves, dimensional analysis, symmetry

Topology, developed at the end of the 19th Century to investigate the subtle interaction of analysis and geometry, is now one of the basic disciplines of mathematics. A working knowledge of the language and concepts of topology is essential in fields as diverse as algebraic number theory and non-linear analysis. This unit develops the basic ideas of topology using the example of metric spaces to illustrate and motivate the general theory. Topics covered include: Metric spaces, convergence, completeness and the Contraction Mapping Theorem; Metric topology, open and closed subsets; Topological spaces, subspaces, product spaces; Continuous mappings and homeomorphisms; Compactness Connectedness Hausdorff spaces and normal spaces. You will learn methods and techniques of proving basic theorems in point-set topology and apply them to other areas of mathematics including basic Hilbert space theory and abstract Fourier series. By doing this unit you will develop solid foundations in the more formal aspects of topology, including knowledge of abstract concepts and how to apply them. Applications include the use of the Contraction Mapping Theorem to solve integral and differential equations.

Note: Students are advised to take MATH2968 before attempting this unit.

This unit of study investigates the modern mathematical theory that was originally developed for the purpose of studying polynomial equations. The philosophy is that it should be possible to factorize any polynomial into a product of linear factors by working over a "large enough" field (such as the field of all complex numbers). Viewed like this, the problem of solving polynomial equations leads naturally to the problem of understanding extensions of fields. This in turn leads into the area of mathematics known as Galois theory.The basic theoretical tool needed for this program is the concept of a ring, which generalizes the concept of a field. The course begins with examples of rings, and associated concepts such as subrings, ring homomorphisms, ideals and quotient rings. These tools are then applied to study quotient rings of polynomial rings. The final part of the course deals with the basics of Galois theory, which gives a way of understanding field extensions.

This unit is an introduction to Differential Geometry, one of the core pillars of modern mathematics. Using ideas from calculus of several variables, we develop the mathematical theory of geometrical objects such as curves, surfaces and their higher-dimensional analogues. Differential geometry also plays an important part in both classical and modern theoretical physics. The unit aims to develop geometrical ideas such as curvature in the context of curves and surfaces in space, leading to the famous Gauss-Bonnet formula relating the curvature and topology of a surface.

Measure theory is the study of such fundamental ideas as length, area, volume, arc length and surface area. It is the basis for the integration theory used in advanced mathematics since it was developed by Henri Lebesgue in about 1900. Moreover, it is the basis for modern probability theory. The course starts by setting up measure theory and integration, establishing important results such as Fubini's Theorem and the Dominated Convergence Theorem which allow us to manipulate integrals. This is then applied to Fourier Analysis, and results such as the Inversion Formula and Plancherel's Theorem are derived. The Radon-Nikodyn Theorem provides a representation of measures in terms of a density. Probability theory is then discussed with topics including distributions and conditional expectation.

This unit of study provides an introduction to fluid dynamics, starting with a description of the governing equations and the simplifications gained by using stream functions or potentials. It develops elementary theorems and tools, including Bernoulli's equation, the role of vorticity, the vorticity equation, Kelvin's circulation theorem, Helmholtz's theorem, and an introduction to the use of tensors. Topics covered include viscous flows, lubrication theory, boundary layers, potential theory, and complex variable methods for 2-D airfoils. The unit concludes with an introduction to hydrodynamic stability theory and the transition to turbulent flow.

This unit provides a comprehensive treatment of dynamical systems using the mathematically sophisticated framework of Lagrange and Hamilton. This formulation of classical mechanics generalizes elegantly to modern theories of relativity and quantum mechanics. The unit develops dynamical theory from the Principle of Least Action using the calculus of variations. Emphasis is placed on the relation between the symmetry and invariance properties of the Lagrangian and Hamiltonian functions and conservation laws. Coordinate and canonical transformations are introduced to make apparently complicated dynamical problems appear very simple. The unit will also explore connections between geometry and different physical theories beyond classical mechanics.Students will be expected to solve fully dynamical systems of some complexity including planetary motion and to investigate stability using perturbation analysis. Hamilton-Jacobi theory will be used to elegantly solve problems ranging from geodesics (shortest path between two points) on curved surfaces to relativistic motion in the vicinity of black holes.This unit is a useful preparation for units in dynamical systems and chaos, and complements units in differential equations, quantum theory and general relativity.

This unit continues the study of functions of a complex variable introduced in the second year unit Analysis (MATH2023/2923). It is aimed at highlighting certain topics from analytic function theory that have wide applications and intrinsic beauty. By learning about the analysis of functions of a complex variable, you will acquire a very important background for mathematical areas such as dynamics, algebraic and differential geometry, and number theory; and advanced theoretical physics such as quantum mechanics, string theory, and quantum field theory. The unit will begin with a revision of properties of complex numbers and complex functions. This will be followed by material on conformal mappings, Riemann surfaces, complex integration, entire and analytic functions, the Riemann mapping theorem, analytic continuation, and Gamma and Zeta functions. Finally, special topics chosen by the lecturer will be presented, which may include elliptic functions, normal families, Julia sets, functions of several complex variables, or complex manifolds. At the end of this unit you will have received a broad introduction and gained a variety of tools to apply them within your further mathematical studies and/or in other disciplines.

Mathematics is ubiquitous in the modern world. Mathematical ideas contribute to philosophy, art, music, economics, business, science, history, medicine and engineering. To really see the power and beauty of mathematics at work, students need to identify and explore interdisciplinary links. Engagement with other disciplines also provides essential foundational skills for using mathematics in the world beyond the lecture room. In this unit you will commence by working on a group project in an area of mathematics that interests you. From this you will acquire skills of teamwork, research, writing and project management as well as disciplinary knowledge. You will then have the opportunity to apply your disciplinary knowledge in an interdisciplinary team to indentify and solve problems and communicate your findings to a diverse audience.